The present invention relates to a method for forming a model of an underground zone, constrained both by dynamic data and static data obtained by means of measurements or observations.
One of the main current research means for improving location techniques relative to zones likely to contain hydrocarbons consists in giving better descriptions of the inner architecture of reservoirs. Engineers therefore essentially have two types of data: dynamic data (reservoir) and static data (seismic).
An inversion technique is commonly used for constructing really representative reservoir models, starting from a geologic model given a priori which is gradually modified in order to reduce the differences between the values of the parameters or physical quantities calculated by the model and values of these quantities that can be observed or measured at certain points of the medium, by best minimizing a cost function J.
The model can be optimized by inversion of static data that can be deduced from seismogram interpretations or that have been obtained by measurements in exploration wells. The most commonly used parameter for inversion is the acoustic impedance of the medium.
The model can also be optimized by inversion of dynamic data: production data spread over a period of time or resulting from well tests. However, the data used concern very localized zones in the immediate neighbourhood of production wells. The parameters used for inversion are here the total permeability of the medium and the porosity thereof In both cases, the inversion algorithm is the algorithm shown in FIG. 1.
The models obtained by inversion of static and dynamic data are achieved separately by specialized reservoir engineering and seismic interpretation teams. The solutions which specialists in these two fields come to are often multiple and generally very different according to whether static data or dynamic data are used.
Consider a direct one-dimensional model relative to the flow of fluids between an injection well and a production well in a reservoir diagrammatically represented (FIG. 2) by a cylinder of length L extending between the two wells. Knowing the pressure Po of the fluid at an initial time, its evolution within a time interval [0,T] is governed by the following equations defining the direct model:                               p          ⁡                      (                          x              ,              0                        )                          =                              p            0                    ⁡                      (            x            )                                              ∀                  x          ∈                      [                          0              ,              L                        ]                                                                        -                          k              ⁡                              (                0                )                                              ⁢                                    ∂              p                                      ∂              x                                ⁢                      (                          0              ,              t                        )                          =                              q            0                    ⁡                      (            t            )                                                                                                  ∀                                  t                  ∈                                            ]                        ⁢            0                    ,          T                ]                                                                    ∂              p                                      ∂              t                                +                                    ∂                              ∂                x                                      ⁢                          (                                                -                                      k                    ⁡                                          (                      x                      )                                                                      ⁢                                                      ∂                    p                                                        ∂                    x                                                              )                                      =        0                                                                                ∀                                  x                  ∈                                            ]                        ⁢            0                    ,                                    L              [                              xe2x80x83                            ⁢                              and                ⁢                                  xe2x80x83                                ⁢                                  ∀                                      t                    ∈                                                              ]                        ⁢            0                    ,          T                ]                                          p          ⁡                      (                          L              ,              t                        )                          =                  p          L                                                                        ∀                              t                ∈                                      ]                    ⁢          0                ,                  T          ]                    
where:
p(x,t) is the pressure of the fluid at the time t and at the position x,
k(x) is the permeability of the rock at the position x,
qo(t) is the rate of inflow at the time t,
PL is the pressure at the distance L from the injection well, that are solved according to a finite-difference numerical scheme. A description of the permeability at the interfaces inspired by the finite-volume schemes allows to take account of the stream continuity at the interfaces. It is assumed that the reservoir is an  less than  less than  assembly of sections  greater than  greater than  of homogeneous lithologies and of different permeabilities to be identified. As pressure measurements have been performed at the injection well, at x=0, the following objective function is constructed:
J(k)=xc2xd∫0T(p(0,t)xe2x88x92{circumflex over (p)}(t))2dt
and inversion of the parameter or of the physical quantity k is thus achieved by means of conventional optimization methods, such as the adjoint state method which allows accurate calculation of the gradient of the cost function in relation to the parameter.
Consider a sedimentary basin (FIG. 3) consisting of several geologic layers, having each their own acoustic impedance, explored by seismic means. Seismograms showing the response of the medium (waves reflected by the subsoil discontinuities) in response to the emission of seismic waves by a source are available. The wave equation is a hyperbolic equation where the unknown y(t,z) is the subsoil vibration depending on the time t and on the depth z, with the acoustic impedance of the medium "sgr"(z) as the parameter. The same type of finte-difference scheme as that used for the flow equation is used to solve this equation, i.e. a scheme using a description of the impedances at the interfaces inspired by the finite-volume schemes. Because of the initial conditions, this is here an explicit scheme. The geologic interfaces located by seismic reflection means are the boundary layers between the media having different acoustic impedances.
The medium being quiescent at the initial time t=0, and knowing the pressure impulse g(t) imparted to the medium at the surface at each time t, vibration y(z,t) is the solution to the direct model given by the following modelling equation system (z being the depth in two-way time):       (          P      s        )    ⁢      {                                                                                        σ                  ⁡                                      (                    z                    )                                                  ⁢                                                                            ∂                      2                                        ⁢                    y                                                        ∂                                          t                      2                                                                                  -                                                ∂                                      ∂                    z                                                  ⁢                                  (                                                            σ                      ⁡                                              (                        z                        )                                                              ⁢                                                                  ∂                        y                                                                    ∂                        z                                                                              )                                                      =            0                                                              for              ⁢                              xe2x80x83                            ⁢              z                         greater than                           0              ⁢                              xe2x80x83                            ⁢              and              ⁢                              xe2x80x83                            ⁢              for              ⁢                                                xe2x80x83                                ⁢                                  xe2x80x83                                            ⁢              t                         greater than             0                                                                          y              ⁡                              (                                  z                  ,                  0                                )                                      =                                                                                ∂                    y                                                        ∂                    t                                                  ⁢                                  (                                      z                    ,                    0                                    )                                            =              0                                                                          for              ⁢                              xe2x80x83                            ⁢              z                        ≥            0                                                                                          -                                  σ                  ⁡                                      (                    0                    )                                                              ⁢                                                ∂                  y                                                  ∂                  z                                            ⁢                              (                                  0                  ,                  t                                )                                      =                          g              ⁡                              (                t                )                                                                                        for              ⁢                              xe2x80x83                            ⁢              t                         greater than             0                                                                          y              ⁡                              (                                  L                  ,                  t                                )                                      =            0                                                              for              ⁢                              xe2x80x83                            ⁢              t                         greater than             0                              
As above, the medium is considered to be an assembly of sections sliced vertically and having each an impedance value. Measurements ŷ being known, a cost function is constructed:
J("sgr")=xc2xd∫0T(y(0, t)xe2x88x92ŷ(t))2dt
which allows to perform the acoustic impedance inversion.
The success of an inversion procedure greatly depends, as it is well-known, on the quality and the quantity of observations available. The more observations, the better the quality of the inversion result. A lack of data leads to averaged information on the parameter to be inverted.
An inverse problem performed on two meshes including information on parameters p1, p2 with an insufficient number of data only allows to obtain the mean value of the two parameters. As for reservoirs for example, it is the harmonic mean {overscore (p)} of the two parameters that will be inverted as the result:       1          p      _        =            1      2        ⁢          (                        1                      p            1                          +                  1                      p            2                              )      
A great number of possible parameter doublets p1 and p2 and consequently of different models for which the (reservoir or seismic) simulations will adjust to the observations corresponds to the same harmonic mean {overscore (p)}. This phenomenon can be verified with the following example of 1D simulation of a flow performed in a reservoir consisting of about twenty meshes, with a permeability ki per mesh (at the centre of the mesh). It is a trimodal medium representing a permeable medium (sandstone for example) crossed by an impermeable barrier (clay for example). One tries to know if the model inversion allows to find the permeability gradient. The data are the pressures recorded at the injection well in the course of time.
As shown in FIG. 4A, the distribution of the permeabilities obtained after inversion (graph k2) with an exact gradient calculated by means of the adjoint state method is very far from the distribution used for creating the injection well data (graph k0), whereas the responses of the direct model (FIG. 4B) applied to this distribution (graph P2) perfectly match the data (graph P0). This shows that several models, sometimes very far from the real model, can solve the inverse problem.
The method according to the invention allows to form, by means of an inversion technique, a representation of the variations, in an underground zone producing fluids, of physical quantities, constrained both by fluid production data and exploration data, these data being obtained by measurements or observations. It comprises:
selecting an a priori geologic structure of the underground zone with a distribution of at least a first physical quantity (k) estimated from exploration data and of at least a second physical quantity ("sgr") estimated from production data,
implementing two initial direct models respectively depending on the two physical quantities, and
optimizing at least one model implemented by iterative application of an inversion of at least one physical quantity, said quantities being related to one another by a combination relation, so as to obtain a representation of the structure of the zone in connection with the physical quantities considered which best meets the combination of exploration and production data.
The method comprises for example iterative optimization of the model implemented by performing a simultaneous inversion of the first physical quantity and of the second physical quantity with optimization of a global cost function depending on these two quantities.
A cost function is for example optimized, which is the sum of terms measuring the differences between the predictions of each direct model and the data obtained by measurement or observation of the corresponding quantity and of difference terms formed by taking account of said combination relation
According to an embodiment, each model implemented is optimized by minimizing the objective function by determination of the exact value of the gradient thereof using the adjoint state method.
Production data are for example measurements obtained in wells through the underground zone and exploration data are seismic signals picked up by seismic receivers in response to waves transmitted in the formation, the first physical quantity is the permeability of the medium and the second physical quantity, the impedance of the medium in relation to these waves.
Production data are for example pressure measurements, exploration data are signals picked up by receivers in response to waves transmitted in the formation (elastic or seismic waves for example), the first parameter is the permeability of the medium and the second parameter is the impedance of the medium in relation to these waves (acoustic impedance for example).
The number of models likely to meet both imposed dynamic and static constraints is much more reduced. Reservoir engineers, by applying the method, therefore obtain improved models concerning reservoir characterization which allow a higher exploitation efficiency. The model obtained for the underground zone allows to better show the permeability or acoustic impedance barriers for example. Zones likely to contain hydrocarbons or zones likely to be used as storage places for gas, waste or other materials can thus be better delimited.